SIAM Journal on Financial Mathematics, Volume 12, Issue 1, Page 110-157, January 2021.
We consider numerical approximations to the quantile hedging price of a European claim in a nonlinear market with Markovian dynamics. We study an equivalent stochastic target problem with the conditional probability of success as a new state variable, in addition to the asset value process. We propose numerical approximations based on piecewise constant policy time stepping coupled with novel finite difference schemes. We prove convergence in the monotone case combining backward stochastic differential equation arguments with the Barles and Jakobsen and Barles and Souganidis approaches for nonlinear PDEs. The difficulties compared to the classical setting consist in the construction of monotone schemes under degeneracy due to the perfectly correlated joint process, the unboundedness of the control variable, and the effect of the boundaries in the probability variable on the analysis. We extend the method to a class of nonmonotone schemes using higher order interpolation and prove convergence for linear drivers. In a numerical section, we illustrate the performance of our schemes by considering an example in a financial market with imperfections, and show that a standard nonmonotone scheme produces financially counterintuitive solutions.

中文翻译：

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分位数对冲问题的数值方案

SIAM 金融数学杂志，第 12 卷，第 1 期，第 110-157 页，2021 年 1 月。
我们考虑在具有马尔可夫动力学的非线性市场中欧洲债权的分位数对冲价格的数值近似。除了资产价值过程之外，我们还研究了一个等效的随机目标问题，将成功的条件概率作为一个新的状态变量。我们提出了基于分段常数策略时间步长和新颖的有限差分方案的数值近似。我们证明了单调情况下的收敛性，将向后随机微分方程参数与 Barles 和 Jakobsen 以及 Barles 和 Souganidis 的非线性 PDE 方法相结合。与经典设置相比的困难在于，由于完全相关的联合过程，控制变量的无界，在退化下构建单调方案，以及概率变量中的边界对分析的影响。我们使用高阶插值将方法扩展到一类非单调方案，并证明线性驱动器的收敛性。在数值部分，我们通过考虑具有不完善性的金融市场中的一个例子来说明我们方案的性能，并表明标准的非单调方案会产生财务上违反直觉的解决方案。